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ABSTRACT: This chapter reviews earthquakes as a complex system. After an introduction (Section 1), this is followed by exploring the idea that complex phenomena often exhibit fractal (power-law) scaling in magntiude, space and time (Section 2). This is followed by a discussion of slider-block models, including idea of chaos, self-organized behaviour and the inverse cascade model (Section 3) and then earthquake hazard assessment (Section 4). This is followed by complexities inherent in intermediate-term earthquake prediction (Section 5), and we conclude with a discussion on the implications of the ideas discussed in this chapter (Section 6).
International Geophysics. 03/2013; 81:209-IV.
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ABSTRACT: In this paper a composite model for earthquake rupture initiation and propagation is proposed. The model includes aspects of damage mechanics, fiber-bundle models, and slider-block models. An array of elements is introduced in analogy to the fibers of a fiber bundle. Time to failure for each element is specified from a Poisson distribution. The hazard rate is assumed to have a power-law dependence on stress. When an element fails it is removed, the stress on a failed element is redistributed uniformly to a specified number of neighboring elements in a given range of interaction. Damage is defined to be the fraction of elements that have failed. Time to failure and modes of rupture propagation are determined as a function of the hazard-rate exponent and the range of interaction. Comment: In Press, Corrected Proof, Available online 13 June 2010, ISSN 0167-8442, (http://www.sciencedirect.com/science/article/B6V55-509B27K-3/2/9a243ee45418da97f1752fce0baa76a9) Keywords: Damage; Fracture; Nucleation; Fiber bundles
07/2010;
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ABSTRACT: Observations at the Mauna Loa Observatory, Hawaii, established the systematic increase of anthropogenic CO2 in the atmosphere. For the same reasons that this site provides excellent globally averaged CO2 data, it may provide temperature data with global significance. Here, we examine hourly temperature records, averaged annually for 1977–2006, to determine linear trends as a function of time of day. For night-time data (22:00 to 06:00, LST (local standard time)) there is a near-uniform warming of 0.040 °C y−1. During the day, the linear trend shows a slight cooling of −0.013 °C y−1 at 12:00 (noon, LST). Overall, at Mauna Loa Observatory, there is a mean warming trend of 0.021 °C y−1. The dominance of night-time warming results in a relatively large annual decrease in the diurnal temperature range (DTR) of −0.050 °C y−1. These trends are consistent with the observed increases in the concentrations of CO2 and its role as a greenhouse gas, and indicate the possible relevance of the Mauna Loa temperature measurements to global warming.
Climate of the Past Discussions. 01/2010;
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ABSTRACT: For the purposes of this study, an interval is the elapsed time between two earthquakes in a designated region; the minimum magnitude for the earthquakes is prescribed. A record-breaking interval is one that is longer (or shorter) than preceding intervals; a starting time must be specified. We consider global earthquakes with magnitudes greater than 5.5 and show that the record-breaking intervals are well estimated by a Poissonian (random) theory. We also consider the aftershocks of the 2004 Parkfield earthquake and show that the record-breaking intervals are approximated by very different statistics. In both cases, we calculate the number of record-breaking intervals (nrb) and the record-breaking interval durations Δtrb as a function of "natural time", the number of elapsed events. We also calculate the ratio of record-breaking long intervals to record-breaking short intervals as a function of time, r(t), which is suggested to be sensitive to trends in noisy time series data. Our data indicate a possible precursory signal to large earthquakes that is consistent with accelerated moment release (AMR) theory.
Nonlinear Processes in Geophysics. 01/2010;
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Bulletin of the Seismological Society of America 01/2010; 100(4):1800-1805. · 1.70 Impact Factor
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ABSTRACT: The ETAS (epidemic type aftershock sequence) model is an empirical
simulation approach to aftershocks statistics. It is based on
Gutenberg-Richter (GR) frequency-magnitude scaling and Omori's law for
the temporal decay of aftershock sequences. In addition an empirical
productivity law is postulated. The BASS (branching aftershock
sequence) model is the self-similar limit of ETAS. The arbitrary
productivity relation in ETAS is replaced by Bath's law that on average
the largest aftershock is a fixed magnitude difference delta{m} (approx
1.2) less than the main shock. A major advantage of the BASS modal is
that the two unconstrained parameters K and alpha; in the ETAS
productivity relation are replaced by delta{m} and the b-value in GR
scaling, both directly constrained by observations. In the ETAS model
Bath's law is not satisfied, it is specified in the BASS model. In the
BASS limit the fraction of main shocks that have foreshocks is
independent of mainshock magnitude, in ETAS the dependence is
exponential which is not confirmed by observations. The primary
criticism of the BASS model is that it can produce infinite numbers of
aftershocks. However, this problem is easily removed by a physically
acceptable limit on the upper magnitudes of aftershocks, an inverse
Bath's law that an aftershock cannot be delta{m} (approx) 3 bigger than
a mainshock.
03/2009; 11:3296.
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ABSTRACT: The branching aftershock sequence (BASS) model is a self-similar statistical model for earthquake aftershock sequences. A prescribed parent earthquake generates a first generation of daughter aftershocks. The magnitudes and times of occurrence of the daughters are obtained from statistical distributions. The first generation daughter aftershocks then become parent earthquakes that generate second generation aftershocks. The process is then extended to higher generations. The key parameter in the BASS model is the magnitude difference Deltam* between the parent earthquake and the largest expected daughter earthquake. In the application of the BASS model to aftershocks Deltam* is positive, the largest expected daughter event is smaller than the parent, and the sequence of events (aftershocks) usually dies out, but an exponential growth in the number of events with time is also possible. In this paper we explore this behavior of the BASS model as Deltam* varies, including when Deltam* is negative and the largest expected daughter event is larger than the parent. The applications of this self-similar branching process to biology and other fields are discussed.
Physical Review E 02/2009; 79(1 Pt 2):016101. · 2.26 Impact Factor
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Geophysical Journal International 01/2009; 152(3):718-728. · 2.42 Impact Factor
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ABSTRACT: Statistical frequency-size (frequency-magnitude) properties of earthquake occurrence play an important role in seismic hazard assessments. The behavior of earthquakes is represented by two different statistics: interoccurrent behavior in a region and recurrent behavior at a given point on a fault (or at a given fault). The interoccurrent frequency-size behavior has been investigated by many authors and generally obeys the power-law Gutenberg-Richter distribution to a good approximation. It is expected that the recurrent frequency-size behavior should obey different statistics. However, this problem has received little attention because historic earthquake sequences do not contain enough events to reconstruct the necessary statistics. To overcome this lack of data, this paper investigates the recurrent frequency-size behavior for several problems. First, the sequences of creep events on a creeping section of the San Andreas fault are investigated. The applicability of the Brownian passage-time, lognormal, and Weibull distributions to the recurrent frequency-size statistics of slip events is tested and the Weibull distribution is found to be the best-fit distribution. To verify this result the behaviors of numerical slider-block and sand-pile models are investigated and the Weibull distribution is confirmed as the applicable distribution for these models as well. Exponents β of the best-fit Weibull distributions for the observed creep event sequences and for the slider-block model are found to have similar values ranging from 1.6 to 2.2 with the corresponding aperiodicities CV of the applied distribution ranging from 0.47 to 0.64. We also note similarities between recurrent time-interval statistics and recurrent frequency-size statistics.
Nonlinear Processes in Geophysics. 01/2009;
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ABSTRACT: The purpose of this paper is to discuss the statistical distributions of recurrence times of earthquakes. Recurrence times
are the time intervals between successive earthquakes at a specified location on a specified fault. Although a number of statistical
distributions have been proposed for recurrence times, we argue in favor of the Weibull distribution. The Weibull distribution
is the only distribution that has a scale-invariant hazard function. We consider three sets of characteristic earthquakes
on the San Andreas fault: (1) The Parkfield earthquakes, (2) the sequence of earthquakes identified by paleoseismic studies
at the Wrightwood site, and (3) an example of a sequence of micro-repeating earthquakes at a site near San Juan Bautista.
In each case we make a comparison with the applicable Weibull distribution. The number of earthquakes in each of these sequences
is too small to make definitive conclusions. To overcome this difficulty we consider a sequence of earthquakes obtained from
a one million year “Virtual California” simulation of San Andreas earthquakes. Very good agreement with a Weibull distribution
is found. We also obtain recurrence statistics for two other model studies. The first is a modified forest-fire model and
the second is a slider-block model. In both cases good agreements with Weibull distributions are obtained. Our conclusion
is that the Weibull distribution is the preferred distribution for estimating the risk of future earthquakes on the San Andreas
fault and elsewhere.
11/2008: pages 777-795;
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ABSTRACT: 1] Aftershock decay is often correlated with the modified Omori's law: dN/dt = t À1 (1 + t/c) Àp , where dN/dt is the occurrence rate of aftershocks with magnitudes greater than a lower cutoff m, t is time since a mainshock, t and c are characteristic times, and p is an exponent. Extending this approach, we derive two possibilities: (1) c is a constant independent of m and t scales with m and (2) c scales with m and t is a constant independent of m. These two are tested by using aftershock sequences of four relatively recent and large earthquakes in Japan. We first determine for each sequence the threshold magnitude above which all aftershocks are completely recorded and use only events above this magnitude. Then, visual inspection of the decay curves and statistical analysis shows that the second possibility is the better approximation for our sequences. This means that the power law decay of smaller aftershocks starts after larger times from the mainshock. We find a close association of our second result with a solution obtained for a damage mechanics model of aftershock decay. The time delays associated with aftershocks, according to the second possibility, can be understood as the times needed to nucleate microcracks (aftershocks). Our result supports the idea that the c value is a real consequence of aftershock dynamics associated with damage evolution.
J. Geophys. Res. 01/2007; 112.
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ABSTRACT: The sandpile, forest-fire and slider-block models are said to exhibit self-organized criticality. Associated natural phenomena include landslides, wildfires, and earthquakes. In all cases the frequency-size distributions are well approximated by power laws (fractals). Another important aspect of both the models and natural phenomena is the statistics of interval times. These statistics are particularly important for earthquakes. For earthquakes it is important to make a distinction between interoccurrence and recurrence times. Interoccurrence times are the interval times between earthquakes on all faults in a region whereas recurrence times are interval times between earthquakes on a single fault or fault segment. In many, but not all cases, interoccurrence time statistics are exponential (Poissonian) and the events occur randomly. However, the distribution of recurrence times are often Weibull to a good approximation. In this paper we study the interval statistics of slip events using a slider-block model. The behavior of this model is sensitive to the stiffness α of the system, α=kC/kL where kC is the spring constant of the connector springs and kL is the spring constant of the loader plate springs. For a soft system (small α) there are no system-wide events and interoccurrence time statistics of the larger events are Poissonian. For a stiff system (large α), system-wide events dominate the energy dissipation and the statistics of the recurrence times between these system-wide events satisfy the Weibull distribution to a good approximation. We argue that this applicability of the Weibull distribution is due to the power-law (scale invariant) behavior of the hazard function, i.e. the probability that the next event will occur at a time t0 after the last event has a power-law dependence on t0. The Weibull distribution is the only distribution that has a scale invariant hazard function. We further show that the onset of system-wide events is a well defined critical point. We find that the number of system-wide events NSWE satisfies the scaling relation NSWE ∝(α-αC)δ where αC is the critical value of the stiffness. The system-wide events represent a new phase for the slider-block system.
Nonlinear Processes in Geophysics. 01/2007;
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ABSTRACT: Virtual California is a topologically realistic simulation of the interacting earthquake faults in California. Inputs to the
model arise from field data, and typically include realistic fault system topologies, realistic long-term slip rates, and
realistic frictional parameters. Outputs from the simulations include synthetic earthquake sequences and space-time patterns
together with associated surface deformation and strain patterns that are similar to those seen in nature. Here we describe
details of the data assimilation procedure we use to construct the fault model and to assign frictional properties. In addition,
by analyzing the statistical physics of the simulations, we can show that that the frictional failure physics, which includes
a simple representation of a dynamic stress intensity factor, leads to self-organization of the statistical dynamics, and
produces empirical statistical distributions (probability density functions: PDFs) that characterize the activity. One type
of distribution that can be constructed from empirical measurements of simulation data are PDFs for recurrence intervals on
selected faults. Inputs to simulation dynamics are based on the use of time-averaged event-frequency data, and outputs include PDFs representing measurements of dynamical variability arising from fault interactions and space-time correlations. As a first step for productively using model-based methods for
earthquake forecasting, we propose that simulations be used to generate the PDFs for recurrence intervals instead of the usual
practice of basing the PDFs on standard forms (Gaussian, Log-Normal, Pareto, Brownian Passage Time, and so forth). Subsequent
development of simulation-based methods should include model enhancement, data assimilation and data mining methods, and analysis
techniques based on statistical physics.
Pure and Applied Geophysics 08/2006; 163(9):1819-1846. · 1.79 Impact Factor
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ABSTRACT: It is well known that earthquakes do not occur randomly in space and time. Foreshocks, aftershocks, precursory activation, and quiescence are just some of the patterns recognized by seismologists. Using the Pattern Informatics technique along with relative intensity analysis, we create a scoring method based on time dependent relative operating characteristic diagrams and show that the occurrences of large earthquakes in California correlate with time intervals where fluctuations in small earthquakes are suppressed relative to the long term average. We estimate a probability of less than 1% that this coincidence is due to random clustering. Furthermore, we show that the methods used to obtain these results may be applicable to other parts of the world.
Nonlinear Processes in Geophysics. 01/2006;
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ABSTRACT: The statistical distribution of recurrence times of characteristic
earthquakes plays an important role in hazard assessment. Assumed
distributions include the exponential (random), Weibull (stretched
exponential), log-normal, and Brownian passage time (inverse Gaussian).
In this paper we argue that the Weibull distribution provides the proper
scaling. This distribution has found wide applicability in statistical
physics. In this paper we present the results of numerical simulations
using a hybrid model that combines the forest-fire model with the
site-percolation model in order to better understand the earthquake
cycle. We consider a square array of sites. At each time step, a "tree"
is dropped on a randomly chosen site and is planted if the site is
unoccupied. When a cluster of "trees" spans the site (a percolating
cluster), all the trees in the cluster are removed ("burned") in a
"fire". The removal of the cluster is analogous to a characteristic
earthquake and planting "trees" is analogous to increasing the regional
stress. We find that the statistical distribution of recurrence times
(number of time steps between model earthquakes) is in much better
agreement with the Weibull distribution than either the log-normal or
Brownian passage time distributions. The coefficient of variation
(aperiodicity) of the distribution is 0.394. We also show that the
synthetic distribution of recurrence times obtained using the "Virtual
California" model is in excellent agreement with the Weibull
distribution. For the Parkfield section the simulated earthquakes have a
coefficient of variation with a value 0.354. The actual earthquakes on
the Parfield section are in good agreement with the Weibull distribution
with a coefficient of variation with a value 0.378.
AGU Fall Meeting Abstracts. 11/2005; -1:07.
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J B Rundle,
P B Rundle,
A Donnellan, D L Turcotte,
R Shcherbakov,
P Li,
B D Malamud,
L B Grant,
G C Fox,
D McLeod,
G Yakovlev,
J Parker,
W Klein,
K F Tiampo
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ABSTRACT: In 1906 the great San Francisco earthquake and fire destroyed much of the city. As we approach the 100-year anniversary of that event, a critical concern is the hazard posed by another such earthquake. In this article, we examine the assumptions presently used to compute the probability of occurrence of these earthquakes. We also present the results of a numerical simulation of interacting faults on the San Andreas system. Called Virtual California, this simulation can be used to compute the times, locations, and magnitudes of simulated earthquakes on the San Andreas fault in the vicinity of San Francisco. Of particular importance are results for the statistical distribution of recurrence times between great earthquakes, results that are difficult or impossible to obtain from a purely field-based approach.
Proceedings of the National Academy of Sciences 11/2005; 102(43):15363-7. · 9.68 Impact Factor
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ABSTRACT: Pattern informatics (PI) technique can be used to detect precursory seismic activation or quiescence and make earthquake forecast. Here we apply the PI method for optimal forecasting of large earthquakes in Japan, using the data catalogue maintained by the Japan Meteorological Agency. The PI method is tested to forecast large (magnitude m >= 5) earthquakes for the time period 1995-2004 in the Kobe region. Visual inspection and statistical testing show that the optimized PI method has forecasting skill, relative to the seismic intensity data often used as a standard null hypothesis. Moreover, we find a retrospective forecast that the 1995 Kobe earthquake (m = 7.2) falls in a seismically anomalous area. Another approach to test the forecasting algorithm is to create a future potential map for large (m >= 5) earthquake events. This is illustrated using the Kobe and Tokyo regions for the forecast period 2000-2009. Based on the resulting Kobe map we point out several forecasted areas: the epicentral area of the 1995 Kobe earthquake, the Wakayama area, the Mie area, and the Aichi area. The Tokyo forecasted map was created prior to the occurrence of the Oct. 23, 2004 Niigata earthquake (m = 6.8) and the principal aftershocks with m >= 5.0. We find that these events occurred in a forecasted area in the Tokyo map. The PI technique for regional seismicity observation substantiates an example showing considerable promise as an intermediate-term earthquake forecasting in Japan. Comment: 36 pages, 6 figures, 1 table
01/2005;
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ABSTRACT: This paper explores log-periodicity in a forest-fire cellular-automata model. At each time step of this model a tree is dropped on a randomly chosen site; if the site is unoccupied, the tree is planted. Then, for a given sparking frequency, matches are dropped on a randomly chosen site; if the site is occupied by a tree, the tree ignites and an 'instantaneous' model fire consumes that tree and all adjacent trees. The resultant frequency-area distribution for the small and medium model fires is a power-law. However, if we consider very small sparking frequencies, the large model fires that span the square grid are dominant, and we find that the peaks in the frequency-area distribution of these large fires satisfy log-periodic scaling to a good approximation. This behavior can be examined using a simple mean-field model, where in time, the density of trees on the grid exponentially approaches unity. This exponential behavior coupled with a periodic or near-periodic sparking frequency also generates a sequence of peaks in the frequency-area distribution of large fires that satisfy log-periodic scaling. We conclude that the forest-fire model might provide a relatively simple explanation for the log-periodic behavior often seen in nature.
Nonlinear Processes in Geophysics. 01/2005;
